© K.Cuthbertson and D.Nitzsche 1 Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Foreign Currency Options

© K.Cuthbertson and D.Nitzsche 2 Contracts and Payoffs Hedging Foreign Currency Receipts using Forwards, Options and Futures Pricing Foreign Currency Options Topics

© K.Cuthbertson and D.Nitzsche 3 Contracts and Payoffs

© K.Cuthbertson and D.Nitzsche 4 Foreign Currency Options Contracts Table : Foreign Currency Options PHSE ContractSize K- Increments Min Price Chge GBP £31,250 $0.0250$ = $3.125 DMDM62,500$ = $6.25 JY JY6,250,000$ = $6.25 Can $CD50,000 $ = $5.00 Hold, TVS 0 = $2m in diversified equity portfolio and ‘

© K.Cuthbertson and D.Nitzsche 5 Fig11.2:Long, Foreign Currency Call STST Profit Strike price K = $140 6 C = 4 $150$144 K = $

© K.Cuthbertson and D.Nitzsche 6  = Max(S T – K, 0) - C = -CifS T < K = S T - K - CifS T > K Break even spot rate is: S T,BE = K + C z op = £31,250 at expiry K = 1.40 $/£ C = 4.0 cents/£ = 0.04 $/£ S T = 1.50($/£) (see figure 11.2): Gross profit = (S T – K) z op = (1.50 – 1.40) £31,250 = $3,125 Invoice price per contract = z op C = £31,250 (0.04($/£)) = $1,250 Net profit:  = (S T - K – C) £31,250 = (1.50($/£) ($/£) ($/£)) £31,250 = (0.06($/£))(£31,250) = $1,875 PROFIT FROM A LONG CALL

© K.Cuthbertson and D.Nitzsche 7 Fig 11.3 : Long, Foreign Currency Put Strike price K = $144 STST Profit 1.50 P = $ $140 K = $

© K.Cuthbertson and D.Nitzsche 8 Profit from Long Put If S T < K140 < 144 Exercise the option (“in-the-money”) Gross profit = ( K - S T ) z = ( ) 31,250 = $1250 Net profit  = (K - S T - P) z = = $ per contract If S T > K Do not exercise the option(out-of-the-money) Loss = (2.5/100) x 31,250 = $ Loss is limited to put premium (insurance)

© K.Cuthbertson and D.Nitzsche 9 Buy (long) call on sterling if you expect sterling to appreciate. Buy long put on sterling if you expect sterling to depreciate Calls and Puts: Speculation

© K.Cuthbertson and D.Nitzsche 10 Hedging Foreign Currency Receipts using Forwards, Options and Futures

© K.Cuthbertson and D.Nitzsche 11 Hedging Foreign Currency (Intuition) US firm makes bid for UK contract, outcome of bid is unknown Receipt of f.c. (GBP) is uncertain FORWARD/FUTURES MARKET Bid successful ~ you are hedged Bid unsuccessful ~ you are not hedged outcome is unfavourable if sterling appreciates ~ have to buy GBP at ‘high’ rate in spot market, to honour delivery in the f.c.

© K.Cuthbertson and D.Nitzsche 12 Hedging Foreign Currency (Intuition) LONG PUT ON GBP Bid successful and GBP appreciates ~ outcome favourable even though put expires worthless, as you sell GBP at ‘high’ rate Bid successful and GBP depreciates ~ outcome ‘favourable’ as you exercise the put and receive K Bid unsuccessful and GBP appreciates ~ loss limited to the put premium, P Bid unsuccessful and GBP depreciates ~ outcome ‘favourable’ as you exercise the put and receive (K-S T ) - P

© K.Cuthbertson and D.Nitzsche 13 US firm :bid for sterling contract, V= £12.5m At F 0 = 1.61($/£), USD equivalent of$20.125m. PUT CONTRACT Strike Price K = 1.60 ($/£) Size of Contract, z p = £31,250 Put Premium P = ($/£) Cost, one Put contract (= z p P) = $ Number of Put Contracts N P = (V/z p ) = (£12.5m / £31,250) = 400 contracts Cost of N p puts = N p (z p P) = VP = $312,500 (Note that V = N p z p ) Hedging Foreign Currency Receipts: Detail

© K.Cuthbertson and D.Nitzsche 14 Possible outcomes S T = 1.65($/£) or S T = 1.50($/£) Bid Successful or Unsuccessful Hedging Foreign Currency Receipts: Detail

© K.Cuthbertson and D.Nitzsche 15 A: Bid Successful (appreciation £) S T = 1.65 No Hedge  = V.S T =(12.5m)1.65= $20.625m Forward Market at F 0 =1.62  = V.F 0 =(12.5m)(1.61) = $20.125m Put Option S T >K, puts not exercised convert £’s at 1.65:  = Spot revenue - Cost of Put V.S T – N p (z p.P)= V ( S T – P ) = 12.5 (1.65 – 0.025) =$20,312,500 Equivalent to Long put + long spot = long call

© K.Cuthbertson and D.Nitzsche 16 B: Bid Successful(depreciation £) S T = 1.50 No Hedge  = V.S T =(£12.5)1.50 = $18.75m Forward Market at F 0 =1.62  = V. F 0 =£12.5(1.61) = $20.125m Put Option: Exercise Puts (locked in K= 160): Payoff from puts+long spot - cost of puts =  = [(K - S T ) + S T ].V – N p (z p P) = K.V – N p (z p P) = 1.60 (12.5m) - $312,500 = $19,687,500 Had you chosen put with K = 161 then the put would have a gross payoff equal to that of the forward.

© K.Cuthbertson and D.Nitzsche 17 C: Bid Unsuccessful (appreciation £) S T = 1.65 No Hedge: No Cash Flow Forward Market at F 0 =1.62 Purchase £12.5m at a cost of S T = 1.65 and receive F 0 =1.61  =(F o – S T ).V == (1.61 – 1.65) £12.5= - $500,000 ( Equivalent to open short position in the F.C. and you are exposed to potential large losses as S increases) Put Option:Not Exercised:(equiv to naked put) Lost put premium = N p (z p P) = V. P = $312,500

© K.Cuthbertson and D.Nitzsche 18 D: Bid Unsuccessful (depreciation £) S T = 1.50 No Hedge: No Cash Flow Forward Market at F 0 =1.61 Purchase £12.5m at a cost of S T = 1.50 and receive F 0 =1.62 on (£12.5m)  =(F o – S T )V == (1.61 – 1.50) £12.5= $1.375m ( Equivalent to open short position in the F.C. and you have potential large gains as S increases) Put Option: Exercise Puts:(equiv to naked put) Purchase, at S T = 1.50 and exercise puts K = 1.60  = (K - S T – P ) V=(1.60– 1.50 –0.025) 12.5 = $937,500

© K.Cuthbertson and D.Nitzsche 19 Bid Successful  = V S 1 + V (F o – F 1 ) = V [F 0 + (S 1 – F 1 )] Bid Unsuccessful  = V (F o – F 1 ) The outcomes are the same as for the forward market if the futures are held to maturity, F 1 = S 1 (and ‘close’, if futures are closed out and basis is small) Hedging: Using Futures

© K.Cuthbertson and D.Nitzsche 20 Pricing Foreign Currency Options

© K.Cuthbertson and D.Nitzsche 21 Pricing Replace q= dividend yield by r f [11.13]C = S N(d 1 ) - K N(d 2 ) [11.14]P = K N(-d 2 ) - S N(-d 1 ) d 1 = d 2 = = d 1 - S is measured as $ per £ (or cents per £),

© K.Cuthbertson and D.Nitzsche 22 Pricing: Alternative Representation [11.16]S = F Substituting [11.16] in [11.13] and [11.14]:: [11.17]C = [F N(d 1 ) - K N(d 2 )] [11.18]P = [K N(-d 2 ) – F N(-d 1 )] d 1 = d 2 =

© K.Cuthbertson and D.Nitzsche 23 Table 11.5: Put-Call Parity: Currency Options Case : S T > KCase : S T < K Portfolio A (1) : Cash S T S T Long Put 0 K-S T Total A S T K Portfolio B (2) : Long Call S T - K 0 US T-bond K K Total A S T K Portfolio A = One long put, plus cash of $A = S 0 invested in a foreign bond Portfolio B = One long call, plus domestic (US) bond of ($) K Returns from Two Portfolios :

© K.Cuthbertson and D.Nitzsche 24 END OF SLIDES